1. Field of the Invention
In general, the present invention relates to the computer-implemented method and apparatus for performing three-dimensional alpha/beta tracking. In particular, the method and apparatus of the present invention utilize a change in acceleration in order to determine a direction of movement along an alpha curve and a beta curve in a probability versus error plane. The change in acceleration of a track is used to introduce or dampen oscillations around the track in order to ensure that a tracking error does not grow exponentially, resulting in unacceptable expansion of its prediction windows, which brings in clutter and undesirable tracks.
2. Description of the Prior Art
In conventional tracking systems, if the probability of detection (P.sub.D) per scan is high, if accurate target location measurements are made, if the target density is low, and if there are only a few false alarms, the design of the correlation logic (i.e., associating detections with tracks) and tracking filter (i.e., filter for smoothing and predicting track positions) is straightforward. However, in a realistic radar environment these assumptions are seldom valid, and the design of an automatic tracking system is complicated. In actual situations, one encounters target fades (changes in signal strength due to multipath propagation, blind speeds, and atmospheric conditions), false alarms (due to noise, clutter, interference, and jamming), and poor radar parameter estimates (due to noise, unstabilized antennas, unresolved targets, target splits, multipath propagation, and propagation effects). An accurate tracking system must deal with all these problems.
Conventional tracking systems include contact entry logic, coordinate systems, tracking filter, maneuver-following logic, track initiation, and correlation logic. The simplest tracking filter is the alpha-beta (.alpha.-.beta.) filter described by EQU x.sub.s (k)=x.sub.p (k)+.alpha.[x.sub.m (k)-x.sub.p (k)] (1) EQU V.sub.s (k)=V.sub.s (k-1)+.beta.[x.sub.m (k)-x.sub.p (k)]/T (2) EQU x.sub.p (k+1)=x.sub.s (k)+V.sub.s (k)T (3)
where x(k) is the smoothed position, V.sub.s (k) is the smoothed velocity, x.sub.p (k) is the predicted position, x.sub.m (k) is the measured position, T is the scanning period (time between detections), and .alpha. and .beta. are the system gains.
The minimal mean-square-error (MSE) filter for performing the tracking when the equation of motion is know as the Kalman filter. The Kalman filter is a popular filter for radar and is a recursive filter which minimizes the MSE. The state equation in xy coordinates for a constant-velocity target is EQU S(t+1)=.theta.(t)+.GAMMA.(t)A(t) (4)
where with X(t) being the state vector at time t, consisting of position and velocity components x(t) , x(t), y(t); t ##EQU1##
+1 being the next observation time; T being the time between observations; and a.sub.x (t) and a.sub.y (t) being random accelerations with covariance matrix Q(t) The observation equation is EQU Y(t)=M(t)+V(t) (5)
where ##EQU2##
with Y(t) being the measurement at time t, consisting of positions x.sub.m (t) and y.sub.m (t), and V(t) being a zero-mean noise whose covariance matrix is R(t).
The problem is solved recursively by first assuming that the problem is solved at time t-1. Specifically, it is assumed that the best estimate X(t-1.vertline.t-1) at time t-1 and its error covariance matrix P(t-1 .vertline.t-1) are known, where the caret in the expression of the form x(t.vertline.s) signifies an estimate and the overall expression signifies that X(t) is being estimated with observations up to Y(s). The six steps involved in the recursive algorithm are:
1. Calculate the one-step prediction EQU X(t.vertline.t-1)=.phi.(t-1)X(t-1.vertline.t-1) (7) PA1 2. Calculate the covariance matrix for the one-step prediction PA1 3. Calculate the predicted observation EQU Y(t.vertline.t-1)=M(t)X(t.vertline.t-1) (9) PA1 4. Calculate the filter gain matrix EQU .DELTA.(t)=P(t.vertline.t-1)M.sup.T (t)[M(t)P(t.vertline.t-1)M.sup.T (t)+R(t)].sup.31 1 (10) PA1 5. Calculate the new smoothed estimate EQU X(t.vertline.t)=X(t.vertline.t-1)+.DELTA.(t)[Y(t)-Y(t.vertline.t-1)] (11) PA1 6. Calculate the new covariance matrix EQU P(t.vertline.t)=[I-.DELTA.(t)M(t)]P(t.vertline.t-1) (12)
P(t.vertline.t-1)=.phi.(t-1)P(t-1.vertline.t-1).phi..sup.T (t-1)+.GAMMA.(t-1)Q(t-1).GAMMA..sup.T (t-1) (8)
In summary, starting with an estimate X(t.vertline.t-1) and its covariance matrix P(t.vertline.t-1), after a new observation Y(t) has been received and the six quantities in the recursive algorithm have been calculated, a new estimate X(t.vertline.t) and its covariance matrix P(t.vertline.t) are obtained.
It has been shown that, for a zero random acceleration Q(t)=O and a constant measurement covariance matrix R(t)=R, the alpha-beta (.alpha.-.beta.) filter can be made equivalent to the Kalman Filter by setting ##EQU3##
and ##EQU4##
on the kth scan. Thus as time passes, .alpha. and .beta. approach zero, applying heavy smoothing to the new samples. Usually it is worthwhile to bound .alpha. and .beta. from zero by assuming a random acceleration Q(t).noteq.O corresponding to approximately a 1-g maneuver.
The predicted position of velocities generated by an alpha/beta tracker can divergently oscillate about a track true position from scan to scan resulting in the loss of a track. These divergent oscillations can be caused by the uncertainties in a track position in velocity, or by linear or angular accelerations in time. Conventional alpha/beta trackers inherently do not fully account for acceleration, if they account for acceleration at all. When these divergent oscillations cyclically deviate from the predictions, the alpha/beta tracker will divergently over- and under-compensate for its errors from scan to scan. In this situation, a small error exponentially grows into a gross error, resulting in an unacceptable expansion of prediction windows, thereby bringing in clutter and undesirable tracks.